There is a dearth of official apps for google's own services on google's own mobile OS.
No official Google Reader App, no official widget.
That may be because they work on optimizing the mobile versions of those web-services, to obviate the need for an additional app. You've already got the browser -- why install an app for a web-service?
This 2008 ...

Google Reader
Google Reader is a Web-based tool for gathering, reading, and sharing all the interesting blogs and websites you read on the web.
It was released by Google on October 7, 2005 and graduated from beta status on September 17, 2007. Is available on any modern web browser with JavaScript enabled.
For further information visit:
Google Support ...

For listening to podcasts, Google has a separate application, called Google Listen, which will sync feeds under the Listen Subscriptions category. Unfortunately, new features are not being actively added, but does provide enough functionality to be useful.

After posting the question here, I contacted all the different developers of google reader apps on the android market that I could find. So far only one developer has answered whose app does exactly what I'm looking for, but 1 is good enough!
https://play.google.com/store/apps/details?id=com.droidlab7.reader.pro
I've tried it out on a Gawker Media RSS ...

On the main screen of the app press the Menu button and select Show all. You should see a list of all your subscriptions below the SUBSCRIPTIONS heading, with unread items showing in bold and the unread count to the right. It should look similar to this:
If you don't see anything listed there, you might want to double check the account you're using with ...

Try installing BetterBatteryStats or GSam Battery Monitor Pro. Both apps can provide you granular data about how your phone's battery is being drained. You know the power hog is Google Reader but these apps can help you determine if the app is killing the battery because it's using a ton of data or if it's initiating long or frequent wake-locks...
If I ...

I've used two such apps, and they both work.
https://market.android.com/details?id=com.newsrob
https://market.android.com/details?id=com.google.android.apps.reader
I suspect that Newsrob will let you save more articles offline, but am not sure.

I would start with gReader, which is a very nice unofficial Google Reader app. It supports offline syncing (and can default to only sync on wifi), and multiple other options to customize it (update intervals, number of items to sync, full page/content only/content with images).
If you like it, there is a paid version gReader Pro which removes advertisements ...

According to the announcement on the Google Reader blog it should be avaiable for phones running donut (Android 1.6) and above, so you're covered for that.
The blog doesn't mention any geographical restrictions and I'm in Europe (UK) and I've got it, so it's definitely available outside the US.
If you've got a barcode reader, then you can just point your ...

According to my market, market differs according to your country, there is no "official google reader" app.
May I suggest downloading NewsRob, it's in the market and there are two versions; paid and free, and even the free version features a widget to launch the application, which is simple and light, and it even has a very good feature, you can make the app ...

It doesn't seem to be possible.
At least not in the post it self. In the list of posts it is however.
The only way I am able to scroll in posts, is sideways, which changes the post.
It could be done programmatically though, by combining and rewriting some stuff. But then it is of course no longer Google Reader :)
A suggestion however (which is what I ...

The syncronization between Google Reader (web) and Google Listen (phone app for listening podcasts) is limited to podcast subscriptions and marking episodes as read/listened and unread/unlistened. There is no direct way to add episodes to your Listen "My listen items" queue from Reader.
You can still access old episodes from Listen by scrolling down and ...

From your Home Screen go to Menu -> Settings -> Applications -> Manage Applications and find gReader. Then click the Clear Cache and Clear Data button. This may lead to you having to sign back in but it will synch with your feeds so you won't really lose anything.

I snatched Read it Later Pro last month during the Christmas promos on the Market. There's also a free version as well, if you want to try it without paying. You can easily download the entries from Google Reader -- select "Share" and then choose "Add to Read it Later" from Google Reader. It will then snatch the whole webpage and even apply some formatting ...

There are a lot of apps in Android Market which offer import from your Google Reader subscriptions. The best of them according to Lifehacker is DoggCatcher Podcast Player. BeyondPod Podcast Manager also offers import from Google Reader. But ACast's description claims that the app has integration with Google Reader, not just import (sorry, can't test it ...

I'd have to double-check on the reality of the sync/import part of the question, but you may want to look at BeyondPod. It claims that it will sync read/unread status, but I've not used that capacity. I've used it for years (on WinMo long before Android was available on a phone) and have loved using it.

I assume you have tried the web-based version. Interestingly enough, if you add a shortcut to it it has a special icon. I've tried some of the other ones and seem to always go back to the web-based version.